7,204 research outputs found
On Affine Reachability Problems
We analyze affine reachability problems in dimensions 1 and 2. We show that the reachability problem for 1-register machines over the integers with affine updates is PSPACE-hard, hence PSPACE-complete, strengthening a result by Finkel et al. that required polynomial updates. Building on recent results on two-dimensional integer matrices, we prove NP-completeness of the mortality problem for 2-dimensional integer matrices with determinants +1 and 0. Motivated by tight connections with 1-dimensional affine reachability problems without control states, we also study the complexity of a number of reachability problems in finitely generated semigroups of 2-dimensional upper-triangular integer matrices
On Nonnegative Integer Matrices and Short Killing Words
Let be a natural number and a set of -matrices
over the nonnegative integers such that the joint spectral radius of
is at most one. We show that if the zero matrix is a product
of matrices in , then there are with . This result has applications in
automata theory and the theory of codes. Specifically, if
is a finite incomplete code, then there exists a word of
length polynomial in such that is not a factor of any
word in . This proves a weak version of Restivo's conjecture.Comment: This version is a journal submission based on a STACS'19 paper. It
extends the conference version as follows. (1) The main result has been
generalized to apply to monoids generated by finite sets whose joint spectral
radius is at most 1. (2) The use of Carpi's theorem is avoided to make the
paper more self-contained. (3) A more precise result is offered on Restivo's
conjecture for finite code
Vector Reachability Problem in
The decision problems on matrices were intensively studied for many decades
as matrix products play an essential role in the representation of various
computational processes. However, many computational problems for matrix
semigroups are inherently difficult to solve even for problems in low
dimensions and most matrix semigroup problems become undecidable in general
starting from dimension three or four.
This paper solves two open problems about the decidability of the vector
reachability problem over a finitely generated semigroup of matrices from
and the point to point reachability (over rational
numbers) for fractional linear transformations, where associated matrices are
from . The approach to solving reachability problems
is based on the characterization of reachability paths between points which is
followed by the translation of numerical problems on matrices into
computational and combinatorial problems on words and formal languages. We also
give a geometric interpretation of reachability paths and extend the
decidability results to matrix products represented by arbitrary labelled
directed graphs. Finally, we will use this technique to prove that a special
case of the scalar reachability problem is decidable
On feedback stabilization of linear switched systems via switching signal control
Motivated by recent applications in control theory, we study the feedback
stabilizability of switched systems, where one is allowed to chose the
switching signal as a function of in order to stabilize the system. We
propose new algorithms and analyze several mathematical features of the problem
which were unnoticed up to now, to our knowledge. We prove complexity results,
(in-)equivalence between various notions of stabilizability, existence of
Lyapunov functions, and provide a case study for a paradigmatic example
introduced by Stanford and Urbano.Comment: 19 pages, 3 figure
Decidability of the Membership Problem for integer matrices
The main result of this paper is the decidability of the membership problem
for nonsingular integer matrices. Namely, we will construct the
first algorithm that for any nonsingular integer matrices
and decides whether belongs to the semigroup generated
by .
Our algorithm relies on a translation of the numerical problem on matrices
into combinatorial problems on words. It also makes use of some algebraical
properties of well-known subgroups of and various
new techniques and constructions that help to limit an infinite number of
possibilities by reducing them to the membership problem for regular languages
Complexity of stability and controllability of elementary hybrid systems
Caption title.Includes bibliographical references (p. 16-18).Supported by ARO. DAAL-03-92-G-0115 Supported by NATO. CRG-961115Vincent D. Blondel, John N. Tsitsiklis
On the complexity of bounded time and precision reachability for piecewise affine systems
Reachability for piecewise affine systems is known to be undecidable,
starting from dimension . In this paper we investigate the exact complexity
of several decidable variants of reachability and control questions for
piecewise affine systems. We show in particular that the region to region
bounded time versions leads to -complete or co--complete problems,
starting from dimension . We also prove that a bounded precision version
leads to -complete problems
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